{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261014-advance-20261110-r03-continuous-image-of-compact-0087",
  "pilot_number": "R0043-22087",
  "number": 22087,
  "family_id": "qf-mod41-compactimg-reverse-implication-boundary",
  "title": {
    "zh-Hant": "第 22087 題：緊緻集的連續像",
    "en": "Question 22087: A Compact Constant Image Does Not Make an Open-Interval Domain Compact"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261014-advance-20261110-r03-continuous-image-of-compact-0087",
    "batch_id": "batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-continuous-image-of-compact"
    ],
    "title_zh": "緊緻集的連續像",
    "stem_zh": "compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-reverse-implication-boundary；資料={\"left\":-13,\"right\":13,\"value\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "(-13, 13, 11, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)",
      "explanation_zh": "Wave6第四層核對=(-13, 13, 11, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)；依完整明示資料獨立重算；對image open cover先拉回K取得finite subcover再推回image；不需injective或surjective；compact image不反推domain compact，preimage compact與image closed皆需額外假設。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "緊緻集的連續像"
      ]
    },
    "variant": {
      "family_id": "qf-mod41-compactimg-reverse-implication-boundary",
      "seed": 202611103087,
      "parameters": {
        "left": -13,
        "right": 13,
        "value": 11,
        "domain": "Finite_topological_compact_image",
        "operation": "compactimg-reverse-implication-boundary",
        "semantic_template": "qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-13,\"right\":13,\"value\":11}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成人工審定、sampled orbit-stabilizer、sampled MVT、reverse compactness、zero-mass division、underspecified recurrence、incomplete divisor inversion、bounded-search PDA rejection、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-14",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261014-advance-20261110-r03-continuous-image-of-compact-0087",
    "batch_id": "batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "8847d5d72731b04791a1acd0ca4d011a586e506cb70b74404e8d23158baf8faf",
      "family_id": "qf-mod41-compactimg-reverse-implication-boundary",
      "seed": 202611103087,
      "semantic_template": "qf-mod41-compactimg-reverse-implication-boundary/{\"left\":-13,\"right\":13,\"value\":11}"
    },
    "fields": {
      "title": "A Compact Constant Image Does Not Make an Open-Interval Domain Compact",
      "stem": "Use the ordinary real open interval (-13,13) with left<right and constant map value=11 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary).",
      "answer": {
        "kind": "expression",
        "value": "(-13, 13, 11, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)",
        "explanation": "The proper open interval is noncompact, the constant map is continuous and its singleton image is compact. The final No refuses the converse. Compact preimages or ambient closed images need separate hypotheses, often Hausdorff for closedness. The exact result, retaining the requested order, is (-13, 13, 11, No, Yes, Yes, No, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)."
      },
      "skills": [
        "A Compact Constant Image Does Not Make an Open-Interval Domain Compact"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "compact K、完整來源與目標拓撲以及total continuous map。操作=compactimg-reverse-implication-boundary；資料={\"left\":-13,\"right\":13,\"value\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use the ordinary real open interval (-13,13) with left<right and constant map value=11 into the real line. Give (left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source operation=compactimg-reverse-implication-boundary;original snapshot and mathematics retained",
      "after": "explicit ordered source output fields=[left, right, constant_value, domain_compact, continuous, image_compact, reverse_implication_claim, boundary]",
      "relation": "implies",
      "rule_id": "source_ordered_interface_explicit_without_javascript",
      "conditions": [],
      "explanation": {
        "zh-Hant": "原題歷史介面標記保留，直接列來源有序欄位，不需JavaScript；MVT由原mode選欄序，不補空欄或看答案形狀猜。tag與固定No保原聲稱範圍。",
        "en": "Retain the historical interface marker and list ordered source fields without JavaScript.MVT fields come from the original mode,without padding or answer-shape inference.Tags and fixed No markers retain their claim scope."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "ORIGINAL WHOLE real interval(-13,13);constant real image value=11",
      "after": "for integers j>=0,relative opens U_j=(L+(R-L)/(j+2),R) cover the interval;ANY finite subfamily is contained in U_max_j and misses a nonempty interval near L,so domain NOT compact",
      "relation": "implies",
      "rule_id": "compactimg_constant_open_interval_counterexample_whole_cover",
      "conditions": [],
      "explanation": {
        "zh-Hant": "任意x>L由Archimedean性選j令門檻低於x，故全族覆蓋；有限族有最大j，取L與其門檻間的點便漏掉。這不是有限取樣或僅靠非closed口號判非compact。",
        "en": "For every x>L,the Archimedean property gives j whose threshold is below x,so the whole family covers.Any finite family has a largest j and misses points between L and its positive threshold.This is an actual cover argument,not finite sampling or a nonclosed slogan."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "ORIGINAL noncompact domain=(-13,13);constant f(x)=11",
      "after": "preimage of ANY target open is either the whole domain or empty,so continuous;image singleton has a one-member subcover from any cover;reverse implication claim=否",
      "relation": "implies",
      "rule_id": "compactimg_constant_map_continuity_singleton_compact_no_converse",
      "conditions": [],
      "explanation": {
        "zh-Hant": "常數函數逆像的兩種情況都open；singleton任意cover中有一個open包含該點即可。故compact像不反推domain compact，也不暗推一般preimage compact或非Hausdorff下像closed。",
        "en": "Both possible preimages of a constant map are open.A cover of a singleton has a member containing its point.Thus a compact image does not force a compact domain,nor general compact preimages or closed images without further hypotheses."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      "-13",
      "13",
      "11",
      false,
      true,
      true,
      false,
      "constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image"
    ]
  },
  "conclusion": {
    "canonical_value": "(-13, 13, 11, 否, 是, 是, 否, constant_continuous_map_from_noncompact_open_interval_has_compact_singleton_image)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations",
      "batch_path": "public/data/questions/batches/batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations/questions.jsonl",
      "batch_sha256": "eaacf2d24560b914960985059a27fe82ab74c756c627b78e147e10bddc70846e",
      "original_line": 87,
      "original_line_sha256": "45e836e235dd1bf9d379d3c1cd7b82cfe75d16eae0346c917ab9f15e3d215f69",
      "question_compact_sha256": "8847d5d72731b04791a1acd0ca4d011a586e506cb70b74404e8d23158baf8faf",
      "english_row_compact_sha256": "dd70bc57f27be4626f1430aa3f30d7606b43773b76abd1e7eab9dd55fff9546e",
      "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261110-r03-continuous-image-of-compact-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-continuous-image-of-compact": {
        "sha256": "033a72b2e9f25c030d4348c2fe05cda62d3477182efb9d2fb0e32b8c1178138c",
        "english_sha256": "86415941d2cdb6419d41013c6d3c1611923949c466f9e07156b1aa77671625b5",
        "original_path": "public/data/mko/continuous-image-of-compact.json",
        "english_path": "public/data/i18n/editions/mko-continuous-image-of-compact.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "747b98e75085a6b55251b2d7e04bbf7c60cb5500e5047439b2798fb5f45538fa",
    "original_candidate_sha256": "8c77f77ff97696744875ae419b23a1225fb61a77525cbb974d02a438ed236cea",
    "original_candidate_line_sha256": "8c77f77ff97696744875ae419b23a1225fb61a77525cbb974d02a438ed236cea"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-009-p44",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "49cf6ec60211c92d1c0c302663af4d44479635dc8181fb61a20c3374288e21c5",
      "question_first_witness_sha256": "10fd25b4643e5ff12753daceaa61f9f2499bf381f65ef96e9d21185c7c0d0c5e"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 3,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "cfcfa8e51a26d1f79f3246c8c4dea09e9c1c63ad018280bb81f427134566c429",
      "arithmetic_steps": 0
    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}